The PDF shape control of the state variable for a class of stochastic systems

Abstract

The shape control of probability density function (PDF) is an important subject in stochastic systems. The PDF-shaping control study has ranged from linear systems to non-linear systems. In this paper we present a PDF-shaping control technique which is useful for a class of non-linear stochastic systems. Controlling the PDF shape requires designing a controller to make the state PDF follow the goal PDF; it is actually to determine the gains of the controller. As we know, the stationary PDF of the state variable is equivalent to the solution of the Fokker–Planck–Kolmogorov (FPK) equation arising from the stochastic system driven with Gaussian white noise. After designing the controller, we derive the solution with some parameters to the corresponding FPK equation, and then solve out the parameters in the solution by the linear-least-squares method, therefore obtaining the gains of the controller. Finally, simulation experiments have been carried out to verify the effectiveness of the approach.

Keywords:

• stochastic systems
• probability density function
• state variable
• shape control
• FPK equation
• linear-least-squares method

Acknowledgements

The authors thank the referees and the editors for their constructive comments to improve the quality of this article. This research was supported by the National Natural Science Foundation of China [grant number 61273127]; the Specialized Research Fund for the Doctoral Program of Higher Education [grant number 20116118110008], [grant number 20106118110009] the Scientific Research Plan Projects of Shannxi Education Department [grant number 12JK0524]; Young Teachers Scientific Research Fund of Xi’an University of Posts and Telecommunications [grant number 110-0434].

Additional information

Notes on contributors

Lingzhi Wang

Lingzhi Wang was born in Henan, China. She received her BS degree in electronic and information engineering and her MS degree in pattern recognition and intelligent systems from Xi’an University of Technology, China, in 2003 and 2006, respectively. She is currently working towards a Ph.D. degree in the School of Automation and Information Engineering, Xi’an University of Technology. Her research interests include optimisation, control, and decision of complicated systems.

Fucai Qian

Fucai Qian was born in Xi’an, China. He received his BE and ME degrees, both in mathematics, and his Ph.D. degree in systems engineering, from Shaanxi Normal University in 1984, Northwest University in 1998, and Xi’an Jiaotong University in 1998, respectively. He was a Postdoctoral Fellow at the Chinese University of Hong Kong in 1999. From 1988 to 1998 he was a lecturer at the Xi’an Petroleum Institute. Since 1999 he has been with the School of Automation and Information Engineering, Xi’an University of Technology, where he is currently a professor. His current research interests include optimal control, stochastic control, non-linear control, and large-scale systems.

Jun Liu

Jun Liu was born in Xi’an, China. He received his BS degree in electric technology from Xi’an Jiaotong University in 1985, his MS degree in electrical automation and his Ph.D. degree in control science and engineering both from Xi’an University of Technology in 1991 and 2006, respectively. Since 1985 he has been with the School of Automation and Information Engineering, Xi’an University of Technology, where he is currently a professor, vice president and supervisor of Ph.D. candidates. His main research interests are intelligent control and moving control systems.